Zeta functions of finite Schreier graphs and their zig zag products

TL;DR

This paper studies the Ihara zeta functions of Schreier graphs from the Basilica group, revealing their covering structures and introducing a generalized replacement product, with implications for zig zag products.

Contribution

It introduces the concept of a generalized replacement product of Schreier graphs and analyzes their covering properties, connecting these to zeta functions and zig zag products.

Findings

$ ext{Gamma}_{n+1}$ is a 2-sheeted unramified normal cover of $ ext{Gamma}_n$.

$ ext{Gamma}_{n+r}$ is a $2^n$-sheeted unramified, non-normal cover of $ ext{Gamma}_r$.

Results extend to zig zag products with a 4-cycle.

Abstract

We investigate the Ihara zeta functions of finite Schreier graphs $\Gamma_n$ of the Basilica group. We show that $\Gamma_{1+n}$ is $2$ sheeted unramified normal covering of $\Gamma_n, ~\forall~ n \geq 1$ with Galois group $\displaystyle \frac{\mathbb{Z}}{2\mathbb{Z}}.$ In fact, for any $n > 1, r \geq 1$ the graph $\Gamma_{n+r}$ is $2^n$ sheeted unramified, non normal covering of $\Gamma_r.$ In order to do this we give the definition of the $generalized$ $replacement$ $product$ of Schreier graphs. We also show the corresponding results in zig zag product of Schreier graphs $\Gamma_n$ with a $4$ cycle.